104 ANALYTICAL MECHANICS If we want to express the fact that kinetic reaction and acceleration are oppositely directed we put the last equation in the vector notation and write, kinetic reaction = — mf. (IV) Equations (I) and (II) and consequently equation (IV) hold good not only when the acceleration is due to a change in the magnitude of the velocity but also when it is due to a change in its direction. As an illustration of this fact consider the following ideal experiment : Let P, Fig. 60, be a particle attached to the end of an inextensible string, which passes through the hole 0, in the middle of the smooth and horizontal table A, and is fastened to the spring balance S. If we project the particle in the plane •of the table in a direction at : right angles to the line OP we will find that it describes a circle .about the point 0, with a speed equal to the speed of projection. We will further observe that ihe balance registers a pull. Now let us examine the forces experienced by the particle -during its motion. The particle is acted upon by three forces, namely, its weight, the reaction of the table, and the pull of the string. Since the surface of the table is perfectly smooth and horizontal the weight and the reaction of the plane exactly balance each other. Therefore the pull of the string is the resultant force. Thus the particle is pulled toward the point 0, but somehow manages to keep the same distance from it; and this in spite of the fact that it is not acted upon by forces which would counterbalance the pull of the string. The explanation is plain. While describing the circle the direction of the velocity of the FIG. 60.